The experiments were conducted on a panel of five cadaveric feet coming from different bodies described in Table 1.

The study was approved by the local Ethics committee (Terre d'Ethique, Saint-Etienne) under the decision IRBN132018/CHUSTE.

When the body arrived, it was directly treated and injected with formaldehyde by the hospital embalmer, then placed in a refrigerator at 4°C.

On average 11 days after the body arrived, it was removed from the refrigerator to separate the lower leg from the upper leg at the knee level; the soft tissue of the lower leg around its upper end is removed, revealing the tibial plateau. The connection between the lower leg so prepared and the compression bench was made by an intermediate mold made of polyurethane resin (PU) cast in a semi-closed cylinder in Polyvinyl Chloride (PVC). This mold links the tibial plateau on one side and a load cell on the other. External anatomical landmarks on the tibial head and the ankle were used in order to align the axis of loading with the axis of the tibia. Once the PU had solidified and therefore the mold was anchored to the anatomical part, the assembly was placed in a freezer at −25°C and remained there for an average of 1 month until 48 h before the tests. Forty-eight hours before the experiments, the foot was placed in a 4°C refrigerator where it remains for 24 h, then was placed at room temperature at 17 degrees for 24 h. Once this defrosting time had elapsed, the foot was attached to the test bench to perform a series of loading tests.

The mechanical loading of the legs was performed in an X-ray tomographic system. A mechanical test bench was developed, its frame being made of two materials: welded steel profiles for parts not exposed to X-rays and treated wood for the columns to reduce X-ray absorption. The load is applied by forcing the axial displacement of the foot in plantar contact with a plane. The displacement is imposed by the axial movement of a threaded shaft. The applied load is continuously recorded using a strain gauge sensor (Phidget 500 Kg S Type Load Cell). The surface plane on which the foot is compressed is adjustable up to 15° in two directions (medial/lateral, antero-posterior) to simulate different foot orientations. The test bench is displayed alone in Figure 2 and placed in the CT scan in Figure 3. The external volume of the bench is 1,500 x 400 x 400 mm³.

The five cadaveric feet underwent a series of loading. Each foot was first placed in its natural flat position, then two 15° angular variations were applied: plantar/dorsal flexion and inversion/eversion. In each configuration, the applied load was continuously recorded and the situation was CT scanned punctually at different loads, with and without wearing a shoe, as described in Table 2.

Three shoes of the same model (Decathlon, Villeneuve-d'Ascq, France) of various sizes were chosen. They are representative of the sports market for occasional runners. The shoes were adapted to each foot size.

A total of 24 scanners, with a resolution of 0.66 x 0.66 x 1.00 mm³, were done for each foot. The data acquired for the bare neutral sequence is displayed in Figure 4.

The skeleton of the foot, composed of all its bones interconnected, was easily segmented from the 3D images of the CT scan by applying a threshold; it produces a volumetric mask called a skeleton.

A classical approach to segment groups of voxels of a threshold, each corresponding to a bone, is to look at their connections and separate the independent groups. Each group can thus be associated to a complete or partial bone part. This simple approach rarely produces good results because the different groups of voxels, each describing a bone, are often interconnected at the joints. Faced with this limitation, the simplest and most manual solution is to remove these voxels creating the interconnections, as no automatic and robust methods are available to our knowledge. Fuzzy logic approaches (Hirano et al., 2000) have not yielded the expected results in our implementation.

To avoid repeating manual operation on all scanners performed on a subject, we used a method based on an optimization process, minimizing the minimum distance between the manually segmented bones in the initial position and the skeleton mask.

The minimization consists in looking at a rigid transformation that aligns two point clouds such as the Iterative Closest Point (ICP) method (Besl and McKay, 1992; Rusinkiewicz and Levoy, 2001). The unconstrained optimization with Sequential Least SQuare Programming minimization (SLSQP) (Kraft, 1988) provides better results here than the singular value decomposition (SVD) of the corresponding closest point covariance matrix between the two sets as used in ICP. The implementation of SLSQP from the SciPy python package (Virtanen et al., 2019) was used.

The problem is formulated as the following nonlinear unconstrained problem written in the Equation (1).

With R the Eulerian angles rotation matrix and T→ the Translation vector, both composing the rigid transformation. pi→, each point of the initial set of bone points composed of N_p points. xi→ the point of the skeleton closest to (R·pi→+T→).

Before individual bone minimization, optimizations of bone segments are performed in a hierarchical fashion in order to obtain acceptable results by avoiding local minima problems. The result of minimization applied to the bone segment defines the initial solution of minimization in the next bone segment. The first treated segment is the entire foot containing all bones except tibia and fibula; its optimization results are presented in Figure 5. The treated segments are described in Table 3. The bone minimization of Metatarse 5 is shown in Figure 6.

The tracking of the middle and distal phalanges on the foot skeleton may be less robust. This can be increased by subtracting the previous bones found in the point cloud describing the entire skeleton during bone-by-bone minimizations. Thus, during the bone-by-bone minimization, the distance will no longer be calculated on the complete skeleton but on remaining points only. Good results have been obtained with bone-by-bone minimization by passing from proximal bones to distal bones starting with the bones of the lower leg, then the ankle, midfoot, metatarsus, proximal, middle, and distal phalanges.

To avoid manual placement of ligaments for each cadaveric foot, the meshes of the generic source model bones, whose coordinates of ligament attachment points are known, were deformed toward the geometries of the patients' target bones. The source model was manually constructed from Visible Human Project (VHP) images and anatomy books.

Aligning the source and target is an essential step to ensure proper morphing progression. A scaling of the source to get closer to the target was performed to reduce the difference between the two meshes. The result of the calcaneus bone alignment is shown in Figure 7A.

Once the geometries were aligned, normal projection and smoothing iterations were applied at the source to approach the target.

Each iteration consisted in the following steps: At time t, compute for each points within the polygonal mesh of the current source X(t)=[x→i(t)], the normalized normal vectors N(t)=[N→i(t)] pointing toward the nearest target surface. Upstream, the nearest distances D→i={di} of the source nodes with the nearest corresponding point to the target were calculated. The index i = 1, …, N_i the number of source nodes.

The next positions of the points were calculated with Equation (2)

With w(t) a weight factor that can evolve through iterations. In our case, we have chosen a linear rise of this factor up to a maximum value k₁ as described in Equation (3).

Good results were obtained with a parameter a equal to 0.025 and w₁ equal to 1. To avoid the summation of mesh quality degradations on all iterations, we gently smooth with windowed-sinc filter (Taubin et al., 1996), implemented in the Visualization Toolkit (VTK) library (Schroeder et al., 2006), the deformed mesh to obtain the result of the next iteration X(t + 1). Iterations were stopped when residual distance was below a threshold. The final iteration of the normal calcaneus projection is shown in Figure 7B.

Once the normal projection is complete, the RBF mesh-morphing can be performed with a variation of the Grassi description (Grassi et al., 2011).

The first step is to establish W(t), the weight matrix with only the value of the landmarks positions. The normal distance measurement (ND) (Kim et al., 2012), of each target point on the normally projected source, provides us with the initial position of the marks p→j, also written p→l and the final position p→j⋆. The index i = 1, …, N_i the number of source nodes; j = l = 1, …, N_j the number of landmark.

With D_land.(t) the landmarks displacements matrix as described in Equation (5) and K_land.(t) the landmarks RBF matrix as defined in Equation (6).

W(t) is computed with the inversion of K_land.(t) as in Equation (7)

Once W(t) was computed, the RBF matrix of the deformed source nodes and landmarks was calculated as equation 8, which is used as an input for the displacement matrix D(t) as described in Equation (9). D(t) is added to the source nodes of the iteration to calculate next node positions as Equation (10)

k is an inverse multiquadratic RBF function defined in 11.

c is the minimum distance of the nearest points between deformed source and the target. β is described as follows Equation (12)

Good results were obtained with a parameter b equal to 10.0 and k₁ equal to 0.1. To avoid cumulating the mesh degradations of each iteration, a slight smoothing, as described above, can be reused to obtain the result of the iteration X(t + 1).

For robust results, 5 morphing iterations were used, increasing the number of landmarks used at each iteration according to their magnitude. The result of calcaneus RBF morphing is shown in Figure 7C.

To register ligament attachment points, displacement vectors between the initial source and the morphed source are interpolated to calculate the position of the attachment points on the patient bone. The attachment points of the ligaments of the calcaneus are shown in the Figure 7D.

We kept three feet out of the five feet tested.

Foot 2276 was discarded because of the considerable difference in load between shod experiments and bare experiments.

Foot 2278 was removed due to a high cadaveric rigidity that prevents foot movement.

The initial scan of the foot 2275, done without load, was performed on the foot in a plantar flexion configuration due to the natural position of this foot, while the feet 2277 and 2279 were initially scanned in the neutral position. The 10% load of body weight (BW) on foot 2279 in plantar flexion could not be achieved due to the length of the test bench being too small for this leg size, forcing a load directly above this ratio.

A greater viscoelastic effect is observed on the bare foot than on the same foot in shoes. Figure 9 shows this difference in viscoelastic effect, especially when the foot is subjected to a load >10% of BW.

The results of bone registration was checked visually and yielded satisfactory positioning. The minimum average distance between each registered bone and the skeleton is 0.28 mm with a standard deviation of 0.09 mm.

The Haursdoff distance between the mesh-morphed bone and the target bone is on average 0.06 mm with a standard deviation of 0.09 mm.

The calculated ligament positions are shown in Figure 10, a visual inspection validates the positioning of the ligaments identical to the interpretation of anatomical atlases.

To first test the interaction between bone translation and footwear; Wilcoxon signed-rank tests were performed on the amplitudes of each bone translation between bare foot and shod foot for each configuration. In 81% of the cases, a significant difference in the amplitude of bone translation was found. The highest differences were observed on dorsal flexion (DF) with a load equal to 0.1 BW. The results are displayed for each foot studied in Figures A1–A4.

The same test is applied to the amplitudes of bone rotation, in 93% of the cases a significant difference is found, again it is in the dorsal flexion configuration with a load equal to 0.1 BW that the greatest difference is observed.

We observed that the difference in bone displacement between shod feet and bare feet, expressed relative to the tibia (see Supplementary Material), are slightly higher for shod feet (mean = 0.04, sd = 0.13). On the contrary, when expressed relative to the calcaneus, which removes the movement of the foot segment, these differences are reversed; the amplitude of the movements of the shod foot is then smaller than that of the bare foot (mean = −1.64 mm, sd = 2.78 mm).

Non-parametric statistical tests of angle amplitudes (see Supplementary Material) were performed to test if wearing the shoe affects the chosen joints. The results are presented in Table 5.

Even though the variation is not statistically significant, the TibTalCal angle expressed in the sagittal plane for ankle plantar-dorsal measurement tends to be higher when wearing shoes (p-value < 0.10); the median of the angular differences between shod and shod feet is 0.89° resulting from a greater dorsal flexion when wearing the shoe.

No significant difference between shod and barefoot for the TibTalCal angle expressed in the coronal plane measuring the varus-valgus of the subtalar joint was found.

The median difference between the shod and bare CalTalSM angles, expressed in the sagittal plane for measuring arch sagging, is −1.14°, which results in less arch sagging when the footwear is used.

The median of the angle difference between CalTalSM with shoe and bare CalTalSM, expressed in the transverse plane for measuring mid-tarsal abduction-adduction, is −4.46°, resulting in less medial displacement of the forefoot over the midfoot when the shoe is used.

Seventy-four ligaments were modeled and their deformation monitored (see Supplementary Material). The impact of the shoe on ligament deformation was assessed ligament by ligament, 49% had a significant difference in strain between the foot in the shoe and the bare foot.

Certain groups of ligaments see their strain reduced or increased by wearing the shoe; this is the case for the following ligaments:

No significant differences were found for the following ligaments: